Finding Triangles and Other Small Subgraphs in Geometric Intersection Graphs
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We consider problems related to finding short cycles, small cliques, small independent sets, and small subgraphs in geometric intersection graphs. We obtain a plethora of new results. For example: * For the intersection graph of n line segments in the plane, we give algorithms to find a 3-cycle in O(n^1.408) time, a size-3 independent set in O(n^1.652) time, a 4-clique in near-O(n^24/13) time, and a k-clique (or any k-vertex induced subgraph) in O(n^0.565k+O(1)) time for any constant k; we can also compute the girth in near-O(n^3/2) time. * For the intersection graph of n axis-aligned boxes in a constant dimension d, we give algorithms to find a 3-cycle in O(n^1.408) time for any d, a 4-clique (or any 4-vertex induced subgraph) in O(n^1.715) time for any d, a size-4 independent set in near-O(n^3/2) time for any d, a size-5 independent set in near-O(n^4/3) time for d=2, and a k-clique (or any k-vertex induced subgraph) in O(n^0.429k+O(1)) time for any d and any constant k. * For the intersection graph of n fat objects in any constant dimension d, we give an algorithm to find any k-vertex (non-induced) subgraph in O(nlog n) time for any constant k, generalizing a result by Kaplan, Klost, Mulzer, Roddity, Seiferth, and Sharir (1999) for 3-cycles in 2D disk graphs. A variety of techniques is used, including geometric range searching, biclique covers, "high-low" tricks, graph degeneracy and separators, and shifted quadtrees. We also prove a near-Ω(n^4/3) conditional lower bound for finding a size-4 independent set for boxes.
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